The geometry of Chazy's homogeneous third-order differential equations

Chazy studied a family of homogeneous third-order autonomous differential equations. They are those, within a certain class, admitting exclusively single-valued solutions. Each one of these equations yields a polynomial vector field in complex three-dimensional space. For almost all of these these vector fields, the Zariski closure of a generic orbit yields an affine surface endowed with a holomorphic vector field that has exclusively single-valued solutions. We classify these surfaces and relate this classification to recent results of Rebelo and the author.